The answer to the riddle is a besom-broom. You can find it in the Scholar's Guild-hall on the second floor of the Last Homely House (not Elrond's Study on the third floor). [Prior to update 13, it was in the Stable.]

Objective 1

Bilbo Baggins: 'I would like to offer one of my poems on Elven lore, but I dread Lindir's gentle laughter more than a goblin's sword! So I will follow his lead and make another homely riddle, one that I am sure you can solve by finding another item here in the Last Homely House. Here it is:

'A long beard I have of bristly hair,

But no chin or cheek on which to bear.

Many hands hold me, with ne'er a caress,

Yet I'm e'er the first to clean up a mess.

Objective 2

Find an item that solves Bilbo's riddle

The solution to Bilbo's riddle is somewhere in the Last Homely House in Rivendell. Bilbo is in the Last Homely House in Rivendell.

Bilbo is waiting for you to bring him the item that solves his riddle.

Bilbo Baggins: 'Ha, has this puzzle stumped you? Perhaps your quick answer to Lindir was just a fluke!'

Lindir: 'Now, now, my friend, it would not be fair for me to give the answer away.'